K-theory and motivic cohomology of singular schemes

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The proposer will study the algebraic K-theory and motivic theories of singular schemes. The first project is to prove Weibel's vanishing conjecture and Vorst's regularity conjecture for negative K-theory in characteristic p; the characteristic 0 case has been previously handled by Cortinas, Haesemeyer, Schlichting and Weibel. The second project deals with Suslin's singular homology. With coefficients prime to the characteristic over an algebraically closed field these groups are dual to etale cohomology by work of Suslin and Voevoedsky. Geisser proposes to study the behavior of its rational and p-primary part in characteristic p (especially over finite fields), as well as the relationship to etale cohomolgy over non-algebraically closed fields.In this project, Geisser will study algebraic K-theory and motivic cohomology theories. These theories form invariants which help to understand the structure of the set of solutions to a system of polynomial equations with coefficients and solutions in fields, like the rational numbers, real numbers or complex numbers. These invariants are fairly well understood if the solution set is smooth (for example, curves are smooth if they have no intersections or cusps).The proposer will try to advance the understanding in the general case, using methods which have been developed recently. This could lead to applications in cryptography, because several cryptosystems rely on (the difficulty in) finding solutions to systems of polynomial equations.